Suppose the box can be filled completely.

Consider the bottom layer of cubes - those that touch the bottom of the box.

Their bottom faces tile the box bottom into squares.

Consider the smallest of these squares.

- It cannot be a corner square:

One of its (larger) neighbors prevents its other (also larger) neighbor from touching it.

- It cannot be an edge square:

Two of its (larger) neighbors prevent its third (also larger) neighbor from touching it.

- It must be an interior square:

Four (larger) neighboring squares can bound a smallest interior square.

Now, the top face of the smallest bottom cube (call it A) and the inner faces of its (larger)

surrounding cubes comprise five walls of a **new space to be filled entirely with cubes.**

Cube A's top face is tiled with squares, the smallest of which is an interior square.The cube (call it B) that sits on this square is surrounded by four larger cubes. Its top face and their inner faces comprise five walls of a **new space to be filled entirely with cubes**.

Cube B's top face is tiled with squares, the smallest of which is an interior square.The cube (call it C) that sits on this square is surrounded by four larger cubes. Its top face and their inner faces comprise five walls of a **new space to be filled entirely with cubes**.

Cube C's top face is tiled with squares, the smallest of which is an interior square.The cube (call it D) that sits on this square is surrounded by four larger cubes. Its top face and their inner faces comprise five walls of a **new space to be filled entirely with cubes**.

... etc. ...

No matter how finely the box bottom is partitioned into squares, or how nearly equal the sizes of the cubes are, there will always be space above the stack of cubes A, B, C, D, ... that remains unfilled.

The box therefore cannot be completely filled with cubes.